Journal 6: Polygons Delia Coloma 9-5

What is a polygon? Sides: 3. Triangle 4. Quadrilateral 5. Pentagon 6. Hexagon 7. Heptagon 8. Octagon 9. Nonagon 10. Decagon 12. Dodecagon A Polygon is a 2-dimensional shape made up of straight lines, and closed. This means that all the lines need to connect .

Found example:

Parts of a polygon: Diagonal Vertex side

Real life examples:

Concave vs. Convex Convex: Concave: It is when a polygon has all vertices pointing outwards. Example 1) Example 2) Example 3) Concave: It is when any figure has 1 or more vertices pointing inwards. Example 1) Example 2) Example 3)

Real life example: Concave: Convex:

Equilateral vs. equiangular It is when all of the sides are congruent. Equiangular: It is when all of the angles are congruent. 1 m 14 cm 2 in

Real Life example: Equiangular: Equilateral:

Then divide by the number of sides.  Interior angles theorem for polygons: If it is a regular polygon: To find each interior angle, you need to use this formula: N-2 x 180 Then divide by the number of sides.

Examples: Hexagon 6-2=4 4x180= 720 720/6=120 quadrilateral 4-2=2 Heptagon 5-2=3 3x180=540 540/5=108 4-2=2 2x180=360 360/4=90

Real life example: 6-2=4 4x180= 720 720/6=120

Theorems of Parallelograms: Opposite sides are congruent Definition of parallelogram: quadrilateral that has opposite sides parallel to each other. Opposite angles are congruent Diagonals bisect each other Consecutive angles are supplementary It has one set of congruent and parallel sides.

Converse of theorems of parallelograms: If it is a parallelogram then opposite sides are congruent quadrilateral that has opposite sides parallel to each other is a parallelogram If their opposite angles are congruent, then it is a parallelogram The diagonals will bisect each other if it is a parallelogram. If the consecutive angles are supplementary then it is a parallelogram If it has one set of congruent and parallel sides, then it is a parallelogram.

How to prove that quadrilateral is a parallelogram… b c d 1 2 3 4 Given: AB is congruent to CD, BC is cong. To DA Prove: ABCD is a parallelogram. statement reason Ab is cong to cd, bc is congr to da given Bd is cong to bd Reflexive property Triangle dab is cong to dcb sss <1 is cong to <3, <4 is cong to <2 cpct Ab || to cd, bc is || to da Abcd is a parallelogram Alternate int. Angles tm Def of parallelogram

Rhombus, squares and rectangles:

Rectangles: It is any parallelogram with 4 right angles. 1. The diagonals bisect each other, so they are congruent. Example 2) Example 1) Example 3)

Real life example:

Rectangle theorems:

Theorem 6-5-1: If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

Theorem 6-5-2: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Rhombus: A parallelogram with 4 congruent sides. 1. Diagonals are perpendicular. Example 1) Example 3) Example 2)

Real life example:

Rhombus theorems:

Theorem 6-5-3: If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

Theorem 6-5-4: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Theorem 6-5-5: If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

Squares: A parallelogram that is both a rectangle and a rhombus. 1) Has parallelogram characteristics.

“Cartoon life example:”

Trapezoids: A quadrilateral with one pair of parallel sides. Base 2 leg Properties of an isosceles trapezoid: Diagonals are congruent Base angles (both sets) are congruent. Opposite angles are supplementary. Isosceles trapezoid: a trapezoid with a pair of congruent legs.

Examples:

Real life example:

Trapezoidal theorems:

Theorem 6-6-3 If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.

Theorem 6-6-4 If a trapezoid has one pair of congruent base angles, then the trapezoid isosceles.

Theorem 6-6-5: A Trapezoid is isosceles if and only if its diagonals are congruent.

Trapezoid midsegment theorem: The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the length of the bases.

Kite: Has two pairs of congruent adjacent sides. Diagonals are perpendicular One pair of congruent angles (the ones formed by non-congruent sides) One of the diagonals bisects the other.

Examples:

Real life example:

Kite theorems:

Theorem 6-6-1: If a quadrilateral is a kite, then its diagonals are perpendicular.

Theorem 6-6-2: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

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